Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
512 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 512 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C23×C89 |
| prime radical | C89×#2733 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C23×C89 ◁ C22×Q8×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#355×C89 | C2 | semidirect |
| 3. C89×#1888 | C2 | semidirect |
| 4. C89×#1857 | C2 | semidirect |
| 5. C89×#1394 | C2 | semidirect |
| 49. #101×C89 | D4 | semidirect |
| 152. #13426 | C89 | direct (N × H) |
| 160. C2×#355 | C2×C89 | semidirect |
| 161. #1888 | C2×C89 | semidirect |
| 162. #1857 | C2×C89 | semidirect |
| 163. #1394 | C2×C89 | semidirect |
| 164. C89 | #13426 | direct (N × H) |
| 208. #101 | D4×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#2733 | 2 |
| C2×#355×C89 | 2 |
| C89×#1888 | 2 |
| C89×#1857 | 2 |
| C89×#1394 | 2 |
| #13426 | 89 |